In this example of a mis-specified model, the non-monotonic relationship between \(\textrm {lpd}\)  and 2-Wasserstein distance means that \(\textrm {lpd}\)  is, at best, a poor proxy of posterior approximation quality.
In fact, we can observe a similar non-monotonic relationship even when the likelihood model is correctly specified (see Appendix ).
One could argue that these examples are not cause for worry in situations where prediction is of primary interest and inference about latent parameters is of secondary concern.
In fact, we will see that conclusions based on lpd do not always align with conclusions based on predictive RMSE below.
Moreover, Proposition 3.6 of [1]} shows that approximation error for predictive distributions closely tracks approximation errors of posterior distributions of latent parameters.
That is, even when one only cares about prediction, an approximate inference algorithm with large \(\textrm {lpd}\)  may still produce large predictive errors because it has not approximated the posterior \(\Pi (\theta \vert \mathcal {D})\)  well.
